Global Existence, Nonexistence And Critical Exponents Of Solutions To Parabolic Systems With Multiple Nonlinearities

This thesis deals with multi-nonlinear parabolic equations (systems). We are interested in the global existence and nonexistence, as well as the critical exponents of solutions. We also give the asymptotic analysis such as blow-up or quenching rate estimates for non-global solutions. It is observed that there may be multi-nonlinearities such as nonlinear diffusion, nonlinear convection, nonlinear reaction (or absorption) and nonlinear boundary flux in complicated parabolic systems. We will introduce some linear algebraic systems containing all the nonlinear exponents in the systems to describe the interactions among all the nonlinearities. We find that there exist essential relationships among the critical exponents p p and the blow-up rates of solutions. It is found that po = PC holds for initial-boundary boundary value problems with bounded domains.The main results obtained in this thesis can be summarized as follows:1. In Chapter 2, we first study single equation problem (I) = A 4| = (x, 0) = X). We get the sufficient and necessary conditions for the global existence and nonexistence, and give the blow-up rate estimate to the solutions. For the more general single equation of the form (II) = A with Neumann boundary condition = g and positive initial value, we establish sufficient condtions for the global existence or nonexistence of solutions.2. In Chapter 3, we will deal with a system of quasilinear parabolic equations with multi-coupled nonlinearities (III) - Au + with boundary flux f = and initial data u(x0) = 0(x), (x0) = V(X). We obtain the sufficient and necessary conditions for the global existence and nonexistence for the case 0 < m,n < 1, and some sufficient conditions for the case m, n . 1 and the case of 0 < m < 1, n 1.3. Chapter 4 is the main part of this thesis. We introduce the so called characteristic algebraic systems to describe the critical exponents for parabolic systems with multi-nonlinearities. We can get simple descriptions for the blow-up criteria and asymptotic behavior of the blowing up solutions by using the critical exponents.iiiThe typical problems considered here are as follows:(IV) = (u) vt = (vn}xx with Neumann boundary conditions - (um)T(0,ii) = and initial data u, (V) ut ?Au ?aiu v = At; ?av with Neumann boundary conditions ?f = rfv02 and initial data u(VI) v with Neumann boundary conditions ;(VII) ut = Au +uv v = AuTM2 + with Dirichlet boundary conditions and initial data u(x,0) .4. In Chapter 5, we study quenching problems:(VIII) ut = V(a(u)Vu) with Neumann boundary condition = g and initial value. We get the quenching rate and asymptotic property of u.(IX) Heat system coupled via negative boundary flux uX(l, t) = - v and u0. We establish simultaneous or non-simultaneous quenching conditions, and quenching rate estimates for u and v.